If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation $x ^ { 2 } + ( 2 - \lambda ) x + ( 10 - \lambda ) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is : (1) $4 \sqrt { 2 }$ (2) 20 (3) $2 \sqrt { 5 }$ (4) $2 \sqrt { 7 }$
If $\tan A$ and $\tan B$ are the roots of the quadratic equation $3 x ^ { 2 } - 10 x - 25 = 0$, then the value of $3 \sin ^ { 2 } ( A + B ) - 10 \sin ( A + B ) \cos ( A + B ) - 25 \cos ^ { 2 } ( A + B )$ is : (1) - 25 (2) 10 (3) - 10 (4) 25
The set of all $\alpha \in R$, for which $w = \frac { 1 + ( 1 - 8 \alpha ) z } { 1 - z }$ is a purely imaginary number, for all $z \in C$ satisfying $| z | = 1$ and $\operatorname { Re } ( z ) \neq 1$, is : (1) $\{ 0 \}$ (2) $\left\{ 0 , \frac { 1 } { 4 } , - \frac { 1 } { 4 } \right\}$ (3) equal to $R$ (4) an empty set
$n$-digit numbers are formed using only three digits 2, 5 and 7. The smallest value of $n$ for which 900 such distinct numbers can be formed is : (1) 9 (2) 7 (3) 8 (4) 6
If $b$ is the first term of an infinite geometric progression whose sum is five, then $b$ lies in the interval (1) $[ 10 , \infty )$ (2) $( - \infty , - 10 ]$ (3) $( - 10,0 )$ (4) $( 0,10 )$
A circle passes through the points $( 2,3 )$ and $( 4,5 )$. If its centre lies on the line $y - 4 x + 3 = 0$, then its radius is equal to : (1) $\sqrt { 5 }$ (2) $\sqrt { 2 }$ (3) 2 (4) 1
Two parabolas with a common vertex and with axes along the $x$-axis and $y$-axis respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is : (1) $3 ( x + y ) + 4 = 0$ (2) $8 ( 2 x + y ) + 3 = 0$ (3) $x + 2 y + 3 = 0$ (4) $4 ( x + y ) + 3 = 0$
If the tangent drawn to the hyperbola $4 y ^ { 2 } = x ^ { 2 } + 1$ intersect the co-ordinates axes at the distinct points $A$ and $B$, then the locus of the midpoint of $AB$ is : (1) $x ^ { 2 } - 4 y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$ (2) $4 x ^ { 2 } - y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$ (3) $x ^ { 2 } - 4 y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$ (4) $4 x ^ { 2 } - y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
The mean of a set of 30 observation is 75. If each observations is multiplied by non-zero number $\lambda$ and then each of them is decreased by 25, their mean remains the same. Then, $\lambda$ is equal to : (1) $\frac { 4 } { 3 }$ (2) $\frac { 1 } { 3 }$ (3) $\frac { 10 } { 3 }$ (4) $\frac { 2 } { 3 }$
An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt { 3 } \mathrm {~km}$ above it is observed at an elevation of $60 ^ { \circ }$ from a point on the ground. If after five seconds, its elevation from the same point is $30 ^ { \circ }$, then the speed (in $\mathrm { km } / \mathrm { hr }$ ) of the aeroplane is (1) 720 (2) 1500 (3) 750 (4) 1440
In a triangle $ABC$, coordinates of $A$ are $(1,2)$ and the equations of the medians through $B$ and $C$ are respectively, $x + y = 5$ and $x = 4$. Then area of $\triangle ABC$ (in sq. units) is : (1) 12 (2) 4 (3) 9 (4) 5
If $f ( x ) = \left| \begin{array} { c c c } \cos x & x & 1 \\ 2 \sin x & x ^ { 2 } & 2 x \\ \tan x & x & 1 \end{array} \right|$, then $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x }$ (1) does not exist (2) exists and is equal to $-2$ (3) exists and is equal to 0 (4) exists and is equal to 2
Let $S$ be the set of all real values of $k$ for which the system of linear equations $x + y + z = 2$ $2 x + y - z = 3$ $3 x + 2 y + k z = 4$ has a unique solution. Then, $S$ is : (1) equal to $R - \{ 0 \}$ (2) an empty set (3) equal to $R$ (4) equal to $\{ 0 \}$
If $x ^ { 2 } + y ^ { 2 } + \sin y = 4$, then the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( - 2,0 )$ is : (1) $-34$ (2) 4 (3) $-2$ (4) $-32$
If a right circular cone, having maximum volume, is inscribed in a sphere of radius $3$ cm, then the curved surface area (in $\mathrm { cm } ^ { 2 }$) of this cone is : (1) $8 \sqrt { 2 } \pi$ (2) $6 \sqrt { 2 } \pi$ (3) $8 \sqrt { 3 } \pi$ (4) $6 \sqrt { 3 } \pi$
If $f \left( \frac { x - 4 } { x + 2 } \right) = 2 x + 1 , ( x \in R - \{ 1 , - 2 \} )$, then $\int f ( x ) d x$ is equal to (1) $12 \ln | 1 - x | - 3 x + C$ (2) $- 12 \ln | 1 - x | - 3 x + C$ (3) $12 \ln | 1 - x | + 3 x + C$ (4) $- 12 \ln | 1 - x | + 3 x + C$
The area (in sq. units) of the region $\{ x \in R : x \geq 0 , y \geq 0 , y \geq x - 2$ and $y \leq \sqrt { x } \}$ is (1) $\frac { 13 } { 3 }$ (2) $\frac { 8 } { 3 }$ (3) $\frac { 10 } { 3 }$ (4) $\frac { 5 } { 3 }$
Q86
First order differential equations (integrating factor)View
Let $y = y ( x )$ be the solution of the differential equation $\frac { d y } { d x } + 2 y = f ( x )$, where $f ( x ) = \left\{ \begin{array} { l l } 1 , & x \in [ 0,1 ] \\ 0 , & \text { otherwise } \end{array} \right.$. If $y ( 0 ) = 0$, then $y \left( \frac { 3 } { 2 } \right)$ is (1) $\frac { e ^ { 2 } - 1 } { e ^ { 3 } }$ (2) $\frac { 1 } { 2 e }$ (3) $\frac { e ^ { 2 } + 1 } { 2 e ^ { 4 } }$ (4) $\frac { e ^ { 2 } - 1 } { 2 e ^ { 3 } }$
A variable plane passes through a fixed point $( 3,2,1 )$ and meets $x , y$ and $z$-axes at $A , B \& C$ respectively. A plane is drawn parallel to the $yz$-plane through $A$, a second plane is drawn parallel to the $zx$-plane through $B$ and a third plane is drawn parallel to the $xy$-plane through $C$. Then the locus of the point of intersection of these three planes, is (1) $\frac { 3 } { x } + \frac { 2 } { y } + \frac { 1 } { z } = 1$ (2) $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = \frac { 11 } { 6 }$ (3) $x + y + z = 6$ (4) $\frac { x } { 3 } + \frac { y } { 2 } + \frac { z } { 1 } = 1$
A box $A$ contains 2 white, 3 red and 2 black balls. Another box $B$ contains 4 white, 2 red and 3 black balls. If two balls are drawn at random, without replacement from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box $B$ is : (1) $\frac { 7 } { 8 }$ (2) $\frac { 9 } { 16 }$ (3) $\frac { 7 } { 16 }$ (4) $\frac { 9 } { 32 }$